A Gray code for arborescences of tournaments

2026-03-30

AI Summary

This study addresses the reconfiguration problem of arborescences rooted at a specified vertex in tournament graphs, aiming to construct a pivot Gray code—an enumeration sequence in which consecutive arborescences differ by exactly one arc. By modeling the problem via a flip graph and analyzing its Hamiltonian properties, the authors establish for the first time the existence of such a Gray code for arborescences in tournaments. Additionally, they identify several sufficient conditions under which the flip graph of arborescences in a general directed graph fails to contain a Hamiltonian cycle. Bridging graph-theoretic reconfiguration, flip graph modeling, and Hamiltonicity analysis, this work provides both theoretical foundations and novel insights for the efficient enumeration of combinatorial structures.

Technology Category

Reasoning under Uncertainty: Graphical ModelsConstraint Satisfaction and Optimization: Other Foundations of Constraint SatisfactionMachine Learning: Probabilistic Circuits and Graphical Models

Application Category

Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphsWeb Mining and Content Analysis: Models for Web evolutionEconomics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We consider the following question of Knuth: given a directed graph $G$ and a root $r$, can the arborescences of $G$ rooted in $r$ be listed such that any two consecutive arborescences differ by only one arc? Such an ordering is called a pivot Gray code and can be formulated as a Hamiltonian path in the reconfiguration graph of the arborescences of $G$ under arc flips, also called flip graph of $G$. We give a positive answer for tournaments and explore several conditions showing that the flip graph of a directed graph may contain no Hamiltonian cycles.

Problem

• Gray code
• arborescence
• tournament
• Hamiltonian path
• flip graph

Innovation

• pivot Gray code
• arborescence
• tournament
• flip graph
• Hamiltonian path